Problem: Divide the following complex numbers: $\dfrac{2(\cos(\frac{11}{12}\pi) + i \sin(\frac{11}{12}\pi))}{\cos(\pi) + i \sin(\pi)}$ (The dividend is plotted in blue and the divisor in plotted in green. Your current answer will be plotted orange.)
Solution: Dividing complex numbers in polar forms can be done by dividing the radii and subtracting the angles. The first number ( $2(\cos(\frac{11}{12}\pi) + i \sin(\frac{11}{12}\pi))$ ) has angle $\frac{11}{12}\pi$ and radius 2. The second number ( $\cos(\pi) + i \sin(\pi)$ ) has angle $1\pi$ and radius 1. The radius of the result will be $\frac{2}{1}$ , which is 2. The difference of the angles is $\frac{11}{12}\pi - 1\pi = -\frac{1}{12}\pi$ The angle $-\frac{1}{12}\pi$ is negative. A complex number goes a full circle if its angle is increased by $2 \pi$ , so it goes back to itself. Because of that, angles of complex numbers are convenient to keep between $0$ and $2 \pi$ $-\frac{1}{12}\pi + 2 \pi = \frac{23}{12}\pi$ The radius of the result is $2$ and the angle of the result is $\frac{23}{12}\pi$.